Question

Дачник привез на рынок корзину яблок. Первому покупателю он продал половину всех яблок и еще пол-яблока, второму – половину остатка и еще пол-яблока и т. д. Когда же пришел шестой покупатель и купил у дачника половину оставшихся яблок и пол-яблока, то оказалось, что у него, как и у всех покупателей, все яблоки были целые и что дачник продал все свои яблоки. Сколько яблок дачник привез на рынок?

Answer 1

Problem Analysis

The problem states that a gardener brought a basket of apples to the market. The gardener sold half of the apples to the first buyer, then half of the remaining apples plus half of an apple to the second buyer, and so on. When the sixth buyer came and bought half of the remaining apples and half of an apple, it turned out that all the buyers had whole apples and the gardener had sold all of his apples. The question is how many apples did the gardener bring to the market?

Solution

Let's solve this problem step by step.

1. The first buyer bought half of the apples and half of an apple. This means that the gardener had twice the number of apples the first buyer bought, plus one apple.

2. The second buyer bought half of the remaining apples and half of an apple. This means that the gardener had twice the number of apples the second buyer bought, plus one apple.

3. We can continue this pattern for each buyer until the sixth buyer.

4. When the sixth buyer came, they bought half of the remaining apples and half of an apple. This means that the gardener had twice the number of apples the sixth buyer bought, plus one apple.

5. At this point, we know that the gardener sold all of his apples. So the number of apples the gardener had when the sixth buyer came is equal to the number of apples the sixth buyer bought, plus one apple.

6. We can use this information to work backward and calculate the number of apples the gardener brought to the market.

Let's calculate the number of apples step by step:

1. The sixth buyer bought half of the remaining apples and half of an apple. So the gardener had twice the number of apples the sixth buyer bought, plus one apple.

2. The fifth buyer bought half of the remaining apples and half of an apple. So the gardener had twice the number of apples the fifth buyer bought, plus one apple.

3. We can continue this pattern until the first buyer.

4. The first buyer bought half of the apples and half of an apple. So the gardener had twice the number of apples the first buyer bought, plus one apple.

Now, let's calculate the number of apples the gardener brought to the market.

Calculation

Let's use a variable to represent the number of apples the gardener brought to the market.

Let's assume the number of apples the gardener brought to the market is x.

1. The first buyer bought half of the apples and half of an apple. So the gardener had twice the number of apples the first buyer bought, plus one apple. - The gardener had 2 * (x/2) + 1 apples after the first buyer.

2. The second buyer bought half of the remaining apples and half of an apple. So the gardener had twice the number of apples the second buyer bought, plus one apple. - The gardener had 2 * (2 * (x/2) + 1)/2 + 1 apples after the second buyer.

3. We can continue this pattern until the sixth buyer. - The gardener had 2 * (2 * (2 * (2 * (2 * (x/2) + 1)/2 + 1)/2 + 1)/2 + 1)/2 + 1 apples after the sixth buyer.

4. At this point, we know that the gardener sold all of his apples. So the number of apples the gardener had when the sixth buyer came is equal to the number of apples the sixth buyer bought, plus one apple. - The gardener had 2 * (2 * (2 * (2 * (2 * (x/2) + 1)/2 + 1)/2 + 1)/2 + 1)/2 + 1 = x/2 + 1 apples when the sixth buyer came.

5. Since the gardener sold all of his apples, the number of apples the gardener had when the sixth buyer came is equal to the number of apples the sixth buyer bought, plus one apple. - x/2 + 1 = x/2 + 1

From this equation, we can see that the number of apples the gardener brought to the market can be any positive number. There is no unique solution.

Therefore, we cannot determine the exact number of apples the gardener brought to the market based on the given information.

Please let me know if there is anything else I can help you with!